Diagonalization of Matrices

This lecture delves into the concept of diagonalizable matrices, exploring the ideal scenario where a matrix is diagonalizable and the process of finding eigenvectors to form a basis. The instructor explains how to complete a set of linearly independent eigenvectors to form a basis, leading to a simpler matrix representation. The lecture also covers the conditions for a matrix to be diagonalizable, including the relationship between algebraic and geometric multiplicities of eigenvalues. Through a concrete example involving population dynamics in a city and countryside, the instructor demonstrates how to analyze the long-term behavior of a system using matrix operations and eigenvectors. The lecture concludes with a discussion on the stability of systems and the practical applications of these mathematical concepts.